Definition:Local Trivialization

Definition

Let $\left({E, M, \pi, F}\right)$ be a fiber bundle.

Let $\operatorname{pr}_1: M \times F \to M$ be the first projection on $M \times F$.

By definition of fiber bundle, for every point $m \in M$ there exists an open neighborhood $U$ of $m$ and a homeomorphism:

$\chi: \pi^{-1} \left({U}\right) \to U \times F$

such that:

$\pi {\restriction}_U = \operatorname{pr}_1 \mathop \circ \chi$

where $\pi {\restriction}_U$ is the restriction of $\pi$ to $U$.

Then the ordered pair $\left({U, \chi}\right)$ is called a local trivialization of $E$ over $U$.